The quadratic spinor Lagrangian, axial torsion current, and generalizations

TL;DR

This paper demonstrates that key gravitational actions can be derived from the Quadratic Spinor Lagrangian using Lounesto's spinor classification, revealing a correspondence between spinor classes and gravity theories, and introduces new boundary and torsion terms.

Contribution

It establishes a one-to-one correspondence between Dirac spinor classes and gravitational actions, and uncovers new boundary and torsion terms in the QSL framework.

Findings

Derived Einstein-Hilbert, Einstein-Palatini, and Holst actions from QSL.

Identified trivial QSL contributions from certain spinor classes.

Discovered new boundary terms and axial torsion currents in specific cases.

Abstract

We show that the Einstein-Hilbert, the Einstein-Palatini, and the Holst actions can be derived from the Quadratic Spinor Lagrangian (QSL), when the three classes of Dirac spinor fields, under Lounesto spinor field classification, are considered. To each one of these classes, there corresponds a unique kind of action for a covariant gravity theory. In other words, it is shown to exist a one-to-one correspondence between the three classes of non-equivalent solutions of the Dirac equation, and Einstein-Hilbert, Einstein-Palatini, and Holst actions. Furthermore, it arises naturally, from Lounesto spinor field classification, that any other class of spinor field (Weyl, Majorana, flagpole, or flag-dipole spinor fields) yields a trivial (zero) QSL, up to a boundary term. To investigate this boundary term we do not impose any constraint on the Dirac spinor field, and consequently we obtain new terms in the boundary component of the QSL. In the particular case of a teleparallel connection, an axial torsion 1-form current density is obtained. New terms are also obtained in the corresponding Hamiltonian formalism. We then discuss how these new terms could shed new light on more general investigations.